Optimal growth when consumption takes time

• Published date: 01 September 2020
• Author: Thi‐Do‐Hanh Nguyen,Cuong Le Van,Thai Ha‐Huy
• Date: 01 September 2020
• DOI: http://doi.org/10.1111/jpet.12467

J Public Econ Theory. 2020;22:1442–1461.wileyonlinelibrary.com/journal/jpet1442

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© 2020 Wiley Periodicals LLC

Received: 18 October 2019

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Accepted: 22 July 2020

DOI: 10.1111/jpet.12467

ORIGINAL ARTICLE

Optimal growth when consumption

takes time

Thai Ha‐Huy

1,2

|Cuong Le Van

3

|Thi‐Do‐Hanh Nguyen

4

1

EPEE, University of Evry, University Paris‐

Saclay, Évry, France

2

TIMAS, Thang Long University, Évry, France

3

IPAG Business School, Paris School of

Economics, CNRS, TIMAS, Paris, France

4

Vietnam Maritime University, 484 Lach

Tray, Hai Phong, Vietnam

Correspondence

Thai Ha‐Huy, EPEE, University of Evry,

University Paris‐Saclay, Évry, France.

Email: thai.hahuy@univ-evry.fr

Funding information

Labex MME‐DII, Grant/Award Number:

ANR‐11‐LBX‐0023‐01

Abstract

This article analyzes a one‐sector growth model

where the consumption takes time. When the con-

sumption takes time, the consumption set is compact

and we meet satiety. However, we prove that dy-

namic constraints are binding. This result is crucial to

prove that, under well‐known assumptions in mac-

roeconomic dynamic programming, the optimal path

is monotonic and always converges to a unique

nontrivial steady state as in the case where con-

sumption is timeless.

1|INTRODUCTION

Many well‐known economic models are based on the assumptions that consumption is in-

stantaneous, that is, it does not take time. In the reality, this assumption is not plausible. In the

past or even today, it took people time to wash clothes or to prepare the meals. Even with the

invention of new machines and technologies, there are activities which cannot be reduced in

time, for example, watching a match of soccer or listening to Beethoven's symphony.

Economic agents must always make the trade‐off between the time devoted for the production,

and the time for leisure. The consumption time constraint was first formally introduced by Gossen

(1983) and much later generalized by Becker (1965) in his famous contribution to the theory of time

allocation. Recently, Tran‐Nam and Pham (2014) consider the question with a general equilibrium

approach. Le Van, Tran‐Nam, Pham, and Nguyen (2018) extend this model to a general equilibrium

model with many goods and many heterogeneous economic agents. In their article, consumption

itself takes time. A typical household is subject to a financial constraint and a time constraint as

well. They show that the economy possesses at least one autarkic Walrasian equilibrium.

In our knowledge, though the problem is well analyzed in static economy, there is no theo-

retical research in a dynamic framework. To give a response for this gap, in this paper, we consider

aone‐sector growth model, in which consumption is itself time consuming. At each period, the

agent must share her/his available amount of time between consuming and working.

In this case, the consumption set is compact, that is, there is a satiation point while in the

usual models where consuming is timeless the consumption is the positive cone and we have no

satiation point. That is the main difference. However, we prove that all the dynamic constraints

are binding in our model. This result is crucial to prove that, under well‐known assumptions in

macroeconomic dynamic programming, such as concavity of the production and utility func-

tions, and complementarity between capital and labor, the optimal path is monotonic and

converges to a unique nontrivial steady state as in the case where consumption is timeless. That

is our main result.

1

We mention that the steady‐state capital stock is lower than the one

obtained in economies where consumption is timeless.

The effects of incorporating an explicit time for consumption in other classes of dynamic

models would be worthwhile to investigate in future work, including for models of pollution

and growth (e.g., Ghosh, Barman, & Gupta, 2020), dynamic allocation of government ex-

penditures (e.g., Fan, Pang, & Pestieau, 2020; Le Van, Nguyen‐Van, Barbier‐Gauchard, &

Le, 2019), or impact of social capital on growth Le Van, Nguyen, Nguyen, and Simioni (2018),

and other dynamic models.

The article is organized as follows. Section 2describes the model. Section 3.2 presents the

indirect utility function and a modified optimization problem which is equivalent to the initial one.

Section 3.4 studies the main properties of the optimal paths. The proofs are given in the appendix.

2|FUNDAMENTALS

Time is discrete. The production requires physical capital and labor. The utility of the agent

depends only on the consumption which takes time. We assume that in each period, the total

amount of time of the agent is inelastic, and is divided into two parts: the first part is devoted for

the production, and the second part for the consumption.

At period

t

, given capital ktand labor which is measured by the labor time

l

t

, the production is

YFkl=(,)

.

ttt

Let

≤≤δ

01

be the depreciation rate of the physical capital. The dynamics of the accumulation

of the capital stocks is

kδkI=(1−)+

,

ttt+1

where Itis the investment at time

t

. It is worth noting that we do not impose that

δ

is strictly

positive.

Given the maximal supply of labor time

L

and the working time

l

t

, the available time for

consumption is Ll−

t

. Hence, there are two constraints for the consumers, the resource con-

straint and the time constraint:

≤

≤

cIY

ac L l

+,

−,

tt t

tt

1

The results of this model are similar to those obtained from a one‐sector growth model with elastic labor, see, for

example, Le Van and Vailakis (2004).

HA‐HUY ET AL.

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